Don't forget that this is also subject to a physical examination of the phone at an Apple store.

I just added AppleCare + to 2 new iPhones today, over the phone. What they did was send a diagnostic to both phones to make sure they were not damaged (assume water damage was what they were checking). The whole process was fairly easy and they didn't require me to bring them in. Nearest Apple Store is about 60+ miles away so that wasn't an option. Just thought I'd mention this, someone might find it useful.